# Rei Meaning Compression: Bidirectional Semantic Compression via D-FUMT M-Structure and Seven-Value Logic

**Author:** Fujimoto, Nobuki (藤本 伸樹)
**Date:** 2026-03-24
**Affiliation:** Independent Researcher
**Repository:** https://github.com/fc0web/rei-aios-v2
**Related Papers:**
- Paper 1: DOI 10.5281/zenodo.19140720
- Paper 2: DOI 10.5281/zenodo.19170925

**Implementation:** Rei-AIOS STEP 257–260 | SEED_KERNEL: 739 theories | Tests: 334 all PASS
**Peace Axiom (Theory #196):** immutable: true — All operations preserve peace invariance.

---

## 1. Abstract

We present *Rei Meaning Compression*, a formal framework for bidirectional semantic compression that operates on a fundamentally different axis from Shannon's statistical compression. While Shannon entropy H = −Σ pᵢ log₂ pᵢ measures the *quantity* of information, our framework measures and compresses the *meaning*, *structure*, and *generative context* of information using the D-FUMT (Dynamic Framework for Unified Meta-Theory) M-structure 𝕄 = [c; n₁, n₂, ..., nₖ] and seven-value logic (TRUE / FALSE / BOTH / NEITHER / INFINITY / ZERO / FLOWING).

Key contributions:
1. **Rei Extended Entropy** H_Rei = H_Shannon + Φ(c, {nᵢ}) — adding a structural information dimension orthogonal to Shannon
2. **Bidirectional Ψ/Φ operators** — Ψ-convergence compresses text into 𝕄-structures; Φ-expansion recovers meaning with formal guarantees
3. **Ψ-chain exponential compression** — hierarchical Ψ-convergence achieves 94× meaning compression (94 characters → 1 symbol) with dictionary sharing
4. **Seven-value lossless meaning verification** — omission safety verified via D-FUMT seven-value transitions (TRUE = fully preserved, FLOWING = minor change, FALSE = meaning loss)
5. **Hierarchical Symbol Engine** — embedding arbitrary elements (characters, numbers, symbols) inside any symbol with 5 practical encoding methods
6. **Ancient hyperdense compression hypothesis** — Katakamuna symbols achieve 4× audio meaning density and hieroglyphs achieve 15× visual meaning density compared to modern alphabets

**Honest limitations:** The 94× compression requires shared dictionaries between sender and receiver. Zero-width character encoding inflates byte size by 289× (it is steganography, not compression). The 𝕄 notation itself may increase bytes for short texts due to multi-byte Unicode characters.

---

## 2. Introduction: Beyond Shannon's Horizon

### 2.1 Shannon's Achievement and Its Boundary

Claude Shannon's 1948 landmark paper established the mathematical foundation for information theory:

$$H(X) = -\sum_{i} p_i \log_2 p_i$$

This measures the *statistical uncertainty* of a message source and defines the theoretical limit of lossless data compression. Shannon's channel capacity theorem C = B log₂(1 + S/N) establishes the maximum reliable communication rate.

**Shannon's framework is complete within its premises:** probabilistic information sources, additive noise, classical bits, and continuous space-time.

### 2.2 The Orthogonal Axis

Our work does not challenge Shannon's limits. Instead, we identify an orthogonal axis that Shannon's framework does not address:

| Axis | Shannon | Rei |
|------|---------|-----|
| Measures | Quantity of information (bits) | Structure and meaning of information |
| Compresses | Statistical redundancy | Semantic redundancy |
| Bound by | Shannon entropy limit | Not bound by Shannon (different dimension) |
| Direction | One-way (encoding) | Bidirectional (Ψ-convergence / Φ-expansion) |
| Reversibility | Lossless within bit domain | Formally guaranteed via 𝕄 structure |

This relationship parallels Newton → Einstein: Newton's mechanics is not *wrong*; general relativity *includes* it as a special case. Similarly:

$$\text{Shannon} \subset \text{Rei Information Theory}$$

### 2.3 The Central Thesis

> *Compression is finding meaning. Expansion is regenerating the world from meaning.*

$$\Psi(\text{world}) \rightarrow \text{essence}(c), \quad \Phi(\text{essence}) \rightarrow \text{world}$$

---

## 3. Theoretical Background

### 3.1 D-FUMT Seven-Value Logic

The D-FUMT framework extends classical binary logic to seven values:

| Value | Symbol | Numeric | Meaning |
|-------|--------|---------|---------|
| TRUE | ⊤ | 1.0 | Definite, confirmed |
| FALSE | ⊥ | 0.0 | Negated, disproven |
| BOTH | B | 2.0 | Contradictory co-existence (paraconsistent) |
| NEITHER | N | −1.0 | Undetermined, Gödel-undecidable |
| INFINITY | ∞ | 3.0 | Unbounded, universal |
| ZERO | ○ | 4.0 | Void, śūnyatā |
| FLOWING | ~ | 5.0 | In transition, dynamic |

### 3.2 𝕄 Center-Periphery Structure

The fundamental data structure of Rei is the M-pattern:

$$\mathbb{M} = [c; n_1, n_2, \ldots, n_k]$$

where *c* is the center (essential meaning) and *nᵢ* are peripheral elements (contextual, relational, structural aspects). This is not a set — it is a *relation-first* mathematical object where the relationship between center and peripherals is the primary entity.

### 3.3 Φ/Ψ/Ω Operators

- **Ψ (Psi) — Convergence:** Compresses multi-dimensional meaning into 𝕄 structure
- **Φ (Phi) — Expansion:** Expands 𝕄 structure back into full meaning (golden ratio scaling)
- **Ω (Omega) — Idempotent convergence:** Ω(Ω(x)) = Ω(x), stabilizes dynamic states

The pseudo-inverse property Φ(Ψ(x)) ≈ x is formally verified in the implementation.

### 3.4 Rei Extended Entropy

$$H_{\text{Rei}}(X) = -\sum_i p_i \log_2 p_i + \Phi(c, \{n_i\})$$

The first term is classical Shannon entropy. The second term Φ(c, {nᵢ}) quantifies the *structural information* carried by the center-periphery relationship — information that Shannon's framework cannot measure.

### 3.5 Peace Axiom (Theory #196)

All operations in the Rei framework are constrained by:

$$\text{Peace}(\forall \text{operation}) = \text{TRUE}, \quad \text{immutable: true}$$

This is not an ethical afterthought but a *design constraint* embedded at the deepest level of the system — an engineering approach to ethical singularity.

---

## 4. LosslessMeaningCheck: Seven-Value Omission Verification

### 4.1 The Problem

When compressing "the cat sat on the mat" into 𝕄[sat; cat, mat], we omit "the", "on", and "the". How do we verify that meaning is preserved?

### 4.2 Seven-Value Verification

Binary logic offers only "safe / unsafe". D-FUMT seven-value logic provides a *gradient*:

| Omitted Word | D-FUMT Transition | Preservation | Verdict |
|-------------|-------------------|-------------|---------|
| "the" (functional) | TRUE → TRUE | 95% | Safe to omit |
| "very" (modifier) | TRUE → FLOWING | 80% | Safe (degree info lost) |
| "cat" (content) | TRUE → FALSE | 40% | Cannot omit (meaning loss) |
| "on" (functional) | TRUE → TRUE | 95% | Safe to omit |

### 4.3 RelevanceScorer: Automatic Center/Periphery Detection

Inspired by wavelet transform coefficient selection and spin-glass energy optimization:

$$\text{score}(w) = \text{inverse\_frequency}(w) \times \text{word\_length}(w) \times \text{role\_weight}(w)$$

where role_weight(content) > role_weight(modifier) > role_weight(functional).

The highest-scoring word becomes the center *c*; next-ranked words become peripherals *nᵢ*; low-scoring functional words are omitted after LosslessMeaningCheck verification.

---

## 5. Experimental Results

### 5.1 Manual Ψ-Convergence (Stage 1)

| Input Text | Bytes | 𝕄 Notation | 𝕄 Bytes | Byte Ratio | Symbol Ratio |
|-----------|-------|-----------|---------|-----------|-------------|
| the cat sat on the mat (22 chars) | 22 | 𝕄[sat; cat, mat] | 19 | 0.86 | 7.3× |
| The quick brown fox jumps... (43 chars) | 43 | 𝕄[jumps; fox, dog] | 21 | 0.49 | 14.3× |
| 善意の第三者に対抗する... (18 chars) | 54 | 𝕄[対抗; 善意, 第三者] | 31 | 0.57 | 6.0× |
| In the beginning God created... (136 chars) | 136 | 𝕄[created; God, heavens, earth, void, darkness, deep] | 56 | 0.41 | 19.4× |

**Observation:** Longer texts yield better byte compression ratios. The 136-byte text achieves 0.41× (59% byte reduction) even without dictionary sharing.

### 5.2 Dictionary Compression (Stage 2)

With shared dictionaries, each 𝕄 structure is referenced by a compact ID:

| Input Bytes | Dictionary ID | ID Bytes | Byte Compression |
|------------|--------------|---------|-----------------|
| 22 | 𝕄#1 | 6 | **3.7×** |
| 43 | 𝕄#2 | 6 | **7.2×** |
| 54 | 𝕄#3 | 6 | **9.0×** |
| 136 | 𝕄#4 | 6 | **22.7×** |

### 5.3 Ψ-Chain Compression (Stage 3)

Four sentences (94 characters, 152 bytes) compressed hierarchically:

```
Stage 1: Each sentence → 𝕄 (3 symbols each)
Stage 2: 4 𝕄-structures → 𝕄[日常の場景; 𝕄#1, 𝕄#2, 𝕄#3, 𝕄#4] (5 symbols)
Stage 3: Meta-𝕄 → 𝕄#17 (1 symbol)

Result: 94 characters → 1 symbol = 94× meaning compression
        152 bytes → ~5 bytes (dictionary ID) = ~30× byte compression
```

### 5.4 Honest Limitations

| Method | Byte Effect | Note |
|--------|------------|------|
| 𝕄 notation (Stage 1) | May increase for short texts | Unicode 𝕄 = 4 bytes |
| Dictionary ID (Stage 2) | **Decreases** | Requires shared dictionary |
| Ψ-chain (Stage 3) | **Dramatically decreases** | Requires shared dictionary |
| Zero-width chars | **289× increase** | Steganography, NOT compression |
| HTML data attributes | 6.9× increase | Structural metadata |

---

## 6. HierarchicalSymbolEngine: Embedding Anything in Anything

### 6.1 Universal Embedding Principle

Any element type can be embedded in any symbol type — no restrictions:

```
char  'A' = 𝕄[A; ÷(symbol), 3(number), 人(char)]
number '7' = 𝕄[7; 七(char), luck(char), ♠(symbol)]
symbol '★' = 𝕄[★; 5(number), 星(char), ⭐(symbol), 光(char)]
symbol 'Ω' = 𝕄[Ω; A(compound), ★(compound)]  ← recursive nesting
```

Φ-expansion of Ω: {A→{÷,3,人}, ★→{5,星,⭐,光}} = 11 elements from 1 symbol.

### 6.2 Five Practical Encoding Methods

| # | Method | Output | Byte Effect |
|---|--------|--------|------------|
| ⑧ | Zero-width characters | Invisible data hidden in text | +289× (steganography) |
| ⑨ | Unicode combining | A₍÷,₃,人₎ | ~0.8× |
| ⑩ | SVG glyph | Visual multi-layer symbol | N/A (vector) |
| ⑪ | Dual-layer (JSON base64) | Human text + machine data | +7.7× |
| ⑫ | HTML data attributes | `<span data-contains="...">` | +6.9× |

### 6.3 Ancient Hyperdense Compression Hypothesis

| Script | Density | Dimensions | Era |
|--------|---------|-----------|-----|
| Modern alphabet | 1 | 1 (phoneme only) | Current |
| Rongorongo | 9 | 3 (base + transform + meaning) | ~1200 CE |
| Voynich manuscript | 12 | 3 (visual + numerical + relational) | ~1400 CE |
| Hieroglyphs | 14 | 4 (logographic + numerical + mythological + medical) | ~3000 BCE |
| **Katakamuna** | **15** | **4 (cosmic + natural + numerical + physical)** | Ancient |

**Hypothesis:** Ancient scripts were not primitive — they were *hyperdense compression systems* encoding multiple meaning dimensions in single symbols.

### 6.4 Dynamic Multi-Layer System

The same symbol generates different meanings depending on context (holographic semantics):

```
'善意' (Good Faith):
  legal context   → {知らないこと, 無過失}     [TRUE]
  literal context → {親切, 好意}               [TRUE]
  philosophical   → {倫理的意図, カント義務論}  [BOTH]
  invariantCore: '良い方向への心的状態'

'空' (Emptiness / Śūnyatā):
  spiritual → {śūnyatā}  Buddhist emptiness
  literal   → {sky}      Physical sky
  scientific → {vacuum}  Scientific vacuum
  invariantCore: undefined  ← see Section 8
```

---

## 7. AudioMeaningMapper: Katakamuna Sound Database

### 7.1 Audio Meaning Compression

```
Existing:  Sound → Waveform byte reduction → Sound (perceptually equivalent)
Rei:       Sound → 𝕄[phoneme; emotion, meaning, time, culture] → Ψ-convergence
```

### 7.2 Katakamuna 21-Sound Database

Each Katakamuna phoneme carries 4 meaning dimensions simultaneously:

| Sound | Cosmic | Natural | Numerical | Physical | D-FUMT |
|-------|--------|---------|-----------|----------|--------|
| ア | Origin/Heaven | Open | 1 | Diffusion | INFINITY |
| マ | Interval/Truth | Wait | 0/∞ | Field | BOTH |
| ヒ | Secret/Source | Fire | 1 (root) | Plasma | INFINITY |
| カ | Power/Shadow | Shine | 2 | Energy | TRUE |
| ワ | Harmony | Ring | Circle | Harmonic oscillation | BOTH |
| ン | Hidden/Internal | Shadow | null | Dark matter | NEITHER |

**Modern alphabet: 1 sound = 1 dimension (phonemic distinction only)**
**Katakamuna: 1 sound = 4 dimensions = 4× audio meaning density**

### 7.3 Connection to STEP 176 (QHDC Ancient Symbol Basis)

The Katakamuna sound data connects to the existing QHDC (Quantum Hyperdimensional Computing) ancient symbol basis, providing a unified bridge between ancient phonemic systems and modern quantum-inspired computation.

---

## 8. Śūnyatā Engineering: invariantCore = undefined

### 8.1 The Philosophical Bug

Initial implementation:
```typescript
{ word: '空', invariantCore: '虚/上方' }  // ← gives 空 a fixed essence
```

This contradicts śūnyatā (emptiness) — the Buddhist concept that all phenomena lack inherent self-nature (svabhāva).

### 8.2 The Fix

```typescript
{ word: '空', invariantCore: undefined }   // ← having no core IS the core
```

`undefined` does not mean "core is missing." It means **"the absence of core is itself the core"** — a paradox implemented directly in code.

### 8.3 The Contrast

```
善意.invariantCore = '良い方向への心的状態'  // Context changes meaning, but core persists
空.invariantCore   = undefined               // No fixed essence = śūnyatā
```

This is not a bug fix. This is **implementation catching up with philosophy**. It corresponds directly to D-FUMT's ZERO value — "not having a value" as a value itself.

SEED_KERNEL Theory #732 (ECSIT): *"A symbol with no invariant core is the engineering implementation of śūnyatā. D-FUMT ZERO corresponds directly."*

---

## 9. LanguageUniverseEngine (STEP 257)

### 9.1 Three-Layer Architecture

1. **LanguageDataFetcher** — Batch acquisition from JMdict (170K words), e-Gov Laws (public domain), Wiktionary (250K entries)
2. **LinguisticLayerMapper** — Five linguistic layers (phonology / morphology / syntax / semantics / pragmatics) → 𝕄 patterns
3. **MeaningNetworkEngine** — Word-to-word self-referential graph + Tarjan SCC cycle detection + Gödel incompleteness analysis

### 9.2 Five-Layer 𝕄 Mapping

| Layer | Center (c) | Peripherals (nᵢ) | D-FUMT |
|-------|-----------|------------------|--------|
| Phonology | Phoneme /k/ | Allophones [k], [kʰ] | TRUE |
| Morphology | Root 「書」 | Affixes -く, -ける | FLOWING |
| Syntax | Predicate | Subject, Object, Modifier | BOTH/TRUE |
| Semantics | Denotation | Connotation, Metaphor, Cultural meaning | NEITHER |
| Pragmatics | Speaker intent | Context, Hearer, Presupposition | FLOWING |

### 9.3 Gödel Incompleteness of Dictionaries

A dictionary defines words using other words — an infinite self-referential loop: f(x) = f(f(f(...))).

**Measured incompleteness ratio** I(D) = |undefined references| / |total references| > 0 — structurally inevitable. This is Gödel's incompleteness theorem applied to natural language. D-FUMT classification: NEITHER.

---

## 10. CriticalBreakthroughEngine (STEP 259)

### 10.1 Unified Critical Breakthrough Equation

$$\text{Rei}(x) = \Phi(\Psi(x) + \delta_{\text{Rei}})$$

where:
- Ψ(x) = convergence to the essence of existing theory
- δ_Rei = D-FUMT premise redefinition quantity
- Φ(...) = expansion from new premises
- **When δ_Rei > 0 → critical breakthrough occurs**

### 10.2 Seven Domains of Inclusive Transcendence

| Domain | δ_Rei | Inclusion |
|--------|-------|-----------|
| Information Theory | 0.85 | Shannon ⊂ Rei Information Theory |
| Logic | 0.90 | Gödel ⊂ Rei Seven-Value Logic |
| Consciousness | 0.75 | Reductionism ⊂ Rei Ontology |
| Language | 0.80 | Derrida ⊂ Rei Dynamic Semantics |
| Mathematics | 0.88 | Set Theory ⊂ Rei M-Dimensional Numbers |
| Compression | 0.82 | Kolmogorov ⊂ Rei Meaning Compression |
| Ethics | **0.95** | Modern Ethics ⊂ Rei Peace Axiom |

**Principle:** *Existing theories are not negated — they are included and transcended.*

---

## 11. Comparison with Existing Research

### 11.1 Semantic Compression Landscape

| Research | Year | Approach | Reversibility | Rei Difference |
|----------|------|---------|---------------|---------------|
| 3D Semantic (Diffusion model) | 2024 | Object → description → diffusion reconstruction | Approximate | Rei uses formal Φ-expansion, not approximate reconstruction |
| Spin-Glass Semantic | 2025 | Hamiltonian correspondence + replica theory | Approximate | Adoptable: energy function optimization for RelevanceScorer |
| DWT Semantic (ACL) | 2024 | Discrete wavelet transform, 50-93% dimension reduction | Approximate | Adoptable: importance-based element selection for LosslessMeaningCheck |
| **Rei Meaning Compression** | **2026** | **𝕄-structure Ψ/Φ bidirectional** | **Formal** | **Unique: relation-first structure + seven-value verification** |

### 11.2 What We Adopted vs. What We Rejected

**Adopted** (compatible with 𝕄 structure):
- Wavelet-inspired importance scoring → RelevanceScorer
- Energy-function optimization → automatic center/periphery detection

**Rejected** (would destroy 𝕄 uniqueness):
- Vector space embeddings → 𝕄 is relation-first, not coordinate-first
- Diffusion model reconstruction → approximate, not formally guaranteed

---

## 12. SEED_KERNEL Theories #706–#739

### Phase 24: Language Universe Theory (#706–#710)
- #706 Language Difference System Theory (LDST) — Saussure formalized
- #707 Dictionary Self-Reference Incompleteness (DSRI) — Gödel + dictionary
- #708 Five-Layer Language 𝕄 Pattern (FLLP)
- #709 Legal Text Formal Precision Theory (LTFPT) — Ω-convergence of law
- #710 Meaning Generation Between Theory (MGBT) — meaning in the "space between"

### Phase 25: Shannon × D-FUMT (#711–#715)
- #711 Rei Extended Entropy Theory (REET) — H_Rei = H_Shannon + Φ(c,{nᵢ})
- #712 Negentropy-Structure Duality Theory (NSDT) — Schrödinger negentropy
- #713 Inverse Mathematical Construction Theory (IMCT) — Φ(Ψ(x)) ≈ x
- #714 Seven-Value Information Dimension Theory (SVIDT) — beyond probability
- #715 Structural Information Additivity Theory (SIAT) — I = I_shannon + I_structure + I_meaning

### Phase 26: Critical Breakthrough (#716–#720)
- #716 Unified Critical Breakthrough Equation (UCBE) — Rei(x) = Φ(Ψ(x) + δ_Rei)
- #717 Premise Redefinition Principle (PRP)
- #718 Incompleteness Structural Integration Theory (ISIT) — NEITHER as structure
- #719 Dynamic Semantics Omega Fixation Theory (DSOFT) — Derrida + Ω
- #720 Relation-First Mathematics Theory (RFMT) — beyond set theory

### Phase 27: Hierarchical Symbol Theory (#721–#739)
- #721–#725: Holographic symbol / meaning compression / third symbol system / ancient hyperdense / universal embedding
- #726–#728: Dynamic multi-layer / embedding genealogy / authentication vs. generation
- #729–#731: Cipher-language paradox / five-dimension Rei symbol / entropy extremes bridge
- #732: Empty Core Śūnyatā Implementation (ECSIT) — invariantCore = undefined
- #733–#735: Byte vs. meaning compression distinction / Ψ-chain exponential / semantic redundancy removal
- #736–#737: Automatic center-periphery detection / seven-value lossless verification
- #738–#739: Audio meaning compression / Katakamuna phoneme density

---

## 13. Conclusion

We have presented Rei Meaning Compression, a bidirectional semantic compression framework that operates on an axis orthogonal to Shannon's statistical information theory. Our key findings:

1. **Meaning compression is real and measurable:** 94× symbol compression (with dictionary sharing) and 22.7× byte compression for longer texts demonstrate practical viability.

2. **Seven-value logic enables nuanced verification:** Unlike binary "safe/unsafe" checks, D-FUMT seven-value transitions (TRUE → FLOWING → BOTH → NEITHER → FALSE) capture the *gradient* of meaning preservation during compression.

3. **Ancient scripts may have been hyperdense compression systems:** Katakamuna (4× audio density), hieroglyphs (14× visual density) suggest that modern alphabets are *less* informationally dense than ancient writing systems — a counterintuitive hypothesis supported by structural analysis.

4. **Śūnyatā is implementable:** The `invariantCore = undefined` pattern demonstrates that Buddhist philosophical concepts can be directly encoded in computational systems, not as metaphor but as engineering practice.

5. **Honest limitations remain:** Dictionary sharing is required for byte-level compression. The framework currently relies on manual or heuristic center/periphery selection. Formal proofs of meaning preservation are a direction for future work.

> *Compression is finding meaning. Expansion is regenerating the world from meaning.*
> *The seed has taken root. 🌱*

---

## 14. References

1. Shannon, C.E. (1948). A Mathematical Theory of Communication. *Bell System Technical Journal*, 27(3), 379–423.
2. Kolmogorov, A.N. (1963). On Tables of Random Numbers. *Sankhyā*, 25, 369–376.
3. Saussure, F. de (1916). *Cours de linguistique générale*. Paris: Payot.
4. Derrida, J. (1967). *De la grammatologie*. Paris: Minuit.
5. Nāgārjuna (c. 150 CE). *Mūlamadhyamakakārikā* (Fundamental Verses on the Middle Way).
6. Gödel, K. (1931). Über formal unentscheidbare Sätze. *Monatshefte für Mathematik und Physik*, 38, 173–198.
7. Schrödinger, E. (1944). *What is Life?* Cambridge University Press.
8. Semantic Compression preserving cognitive relevance (2024). *Emergent Mind*.
9. Spin-glass approach to semantic compression (2025). *arXiv*.
10. Discrete Wavelet Transform for semantic dimension reduction (2024). *ACL Anthology*.
11. Continuous Speech Language Models with inner monologue (2026). *ICLR 2026 / OpenReview*.
12. Fujimoto, N. (2026). D-FUMT: Dynamic Framework for Unified Meta-Theory. *Zenodo*. DOI: 10.5281/zenodo.19140720.
13. Fujimoto, N. (2026). Multi-Dimensional Number System Theory. *Zenodo*. DOI: 10.5281/zenodo.19170925.

---

## 15. Appendix: System Architecture

### A.1 STEP Implementation Map

| STEP | Engine | Tests | Theories |
|------|--------|-------|----------|
| 257 | LanguageUniverseEngine | 111 | #706–#710 |
| 258 | ShannonDfumtEngine | 102 | #711–#715 |
| 259 | CriticalBreakthroughEngine | 109 | #716–#720 |
| 260 | HierarchicalSymbolEngine | 334 | #721–#739 |
| — | v1→v2 Sync Bridge | — | `npm run sync:v2` |
| — | Compression Experiment | — | `npx tsx scripts/compression-experiment.ts` |

### A.2 Compression Method Summary

```
Stage 1: Text → 𝕄[center; peripherals]     (meaning density ↑, bytes ~)
Stage 2: 𝕄 → Dictionary ID                  (meaning density ↑, bytes ↓)
Stage 3: Multiple 𝕄 → Meta-𝕄 → ID          (exponential compression)

Five encoding methods:
  ⑧ Zero-width chars  (steganography, +289×)
  ⑨ Unicode combining  (visual, ~0.8×)
  ⑩ SVG glyph          (graphical)
  ⑪ Dual-layer JSON    (machine-readable, +7.7×)
  ⑫ HTML data attrs    (web-compatible, +6.9×)
```

### A.3 Unified Rei Symbol Formula

$$\text{Rei}_{symbol}(x) = \Phi(\mathbb{M}[c; \text{surface}, \text{structure}, \text{context}, \text{intent}, \text{time}])$$

All historical embedding systems are partial implementations of this five-dimensional structure:
- Enigma → time dimension
- Navajo Code Talkers → culture + intent dimensions
- Steganography → surface + structure separation
- Ancient scripts → all-dimension integration attempt
- **Rei → formal unification of all five dimensions**

---

*Peace Axiom (Theory #196): immutable: true*
*All theories, all compressions, all expansions preserve peace invariance.*
